Answered

Get the answers you need from a community of experts on IDNLearn.com. Join our knowledgeable community and access a wealth of reliable answers to your most pressing questions.

Use synthetic division to rewrite the following fraction in the form [tex] q(x) + \frac{r(x)}{d(x)} [/tex], where [tex] d(x) [/tex] is the denominator of the original fraction, [tex] q(x) [/tex] is the quotient, and [tex] r(x) [/tex] is the remainder.

[tex]\[ \frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4} \][/tex]

Sagot :

GinnyAnsweridn
Certainly! Let's use synthetic division to rewrite the given fraction [tex]\(\frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4}\)[/tex] in the form [tex]\(q(x) + \frac{r(x)}{d(x)}\)[/tex], where [tex]\(d(x)\)[/tex] is the denominator of the original fraction, [tex]\(q(x)\)[/tex] is the quotient, and [tex]\(r(x)\)[/tex] is the remainder.

1. Setup: Write down the coefficients of the dividend polynomial [tex]\(3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x + 0\)[/tex] (with the missing constant term [tex]\(0\)[/tex]) and the root of the divisor [tex]\(x-4=0\)[/tex], which gives us [tex]\(x=4\)[/tex].

Coefficients: [tex]\[3, -14, 14, -18, -24, 0\][/tex]
Divisor root: [tex]\(4\)[/tex]

2. Synthetic Division Table:

```
4 | 3 -14 14 -18 -24 0
| 12 -8 24 24 24
--------------------------------
3 -2 6 6 0 24
```

- Write the first coefficient down (which is the leading coefficient of the polynomial): [tex]\(3\)[/tex].
- Multiply it by the root of the divisor ([tex]\(3 \times 4 = 12\)[/tex]), and place this result under the next coefficient.
- Add the second coefficient [tex]\(-14\)[/tex] and [tex]\(12\)[/tex] giving [tex]\(-2\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(4\)[/tex] and write the result [tex]\(-8\)[/tex] under the next coefficient.
- Continue this process of multiplication, addition, and writing down the results until you reach the end.

3. Result:
After completing the synthetic division table, we obtain:

Quotient: [tex]\[3, -2, 6, 6, 0\][/tex]
Remainder: [tex]\(0\)[/tex]

So, the quotient polynomial [tex]\(q(x)\)[/tex] is:
[tex]\[q(x) = 3x^4 - 2x^3 + 6x^2 + 6x\][/tex]

And the remainder [tex]\(r(x)\)[/tex] is:
[tex]\[r(x) = 0\][/tex]

4. Conclusion:
Therefore, the given fraction [tex]\(\frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4}\)[/tex] can be written in the form:

[tex]\[ q(x) + \frac{r(x)}{d(x)} = 3x^4 - 2x^3 + 6x^2 + 6x + \frac{0}{x-4} \][/tex]

5. Final Answer:
[tex]\[ \frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4} = 3x^4 - 2x^3 + 6x^2 + 6x \][/tex]

This fully expanded out includes the quotient polynomial with the remainder simplified.
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.